**Double Pendulum**:
Javascript pendulum animation Oliver Knill, July 15, 2015.
Reload the page to start with a different initial condition.

x'= (3v-3cos(x-y)w)/(16-9cosSince volume preserving flows on 3 manifolds are expected to be chaotic in general, it is believed that the metric entropy is positive at least for large enough energies. A mathematical proof appears currently completely out of reach. We just don't have the math yet to analyze this. For small energies, KAM (Kolmogorov-Arnold-Moser) dominates and make part of the phase space integrable. There are horse shoes for large energies but this only establishes positive topological entropy. I had tried to apply pluri-subharmonic techniques (developed by Michael Herman) already as an undergraduate at ETH Zürich to attack the metric entropy problem for this problem: the manifold contains a torus which bounds a poly-disc D x D. Now extend this analytically. Since entropy depends in a pluri-subharmonic manner on the larger space using complex parameters (z,w), one can try to estimate the functional at the origin (0,0) of the polydisk, where it does not correspond to a mechanical problem any more but where we would get the estimation job done. Such attempts failed even for much simpler problems, like the Standard map, on which I worked for 15 years and where much more tools from solid state physics and quantum mechanics are available.^{2}(x-y)) y'= (8w-3cos(x-y)v)/(16-9cos^{2}(x-y)) v'=-3x'y' sin(x-y)-9sin(x) w'= 3x'y' sin(x-y)-3sin(y)